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A New Finite Element Gradient Recovery Method:
Superconvergence Property
Zhimin Zhang∗†and Ahmed Naga ‡
Department of Mathematics, Wayne State University, Detroit, MI 48202
Abstract
This is the first in a series of papers where a new gradient recovery method is introduced and analyzed. Itis proved that the method is superconvergent for translation invariant finite element spaces of any order.The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz-Zhupatch recovery method. In addition, under uniform triangular meshes, the method is superconvergent forthe Chevron pattern, and ultraconvergent at element edge centers for the regular pattern. Applicationsof this new gradient recovery technique will be discussed in forthcoming papers.
Key Words. finite element method, least-squares fitting, ZZ patch recovery, superconvergence,ultraconvergence
AMS Subject Classification. 65N30, 65N15, 65N12, 65D10, 74S05, 41A10, 41A25
1 Introduction
A decade has passed since the first appearance of the Zienkiewicz-Zhu gradient patch recovery
method [23] based on a local discrete least-squares fittings. The method is now widely used
in engineering practices for its robustness in a posteriori error estimates and its efficiency in
computer implementation. It is a common belief that the robustness of the ZZ patch recovery
is rooted in its superconvergence property under structured meshes. Even for an unstructured
mesh, when adaptive is used, a mesh refinement will usually bring in some kind of structure
locally. Superconvergence properties of the ZZ patch recovery are proved in [21] for all popular
elements under rectangular mesh and in [10] for the linear element under strongly regular
triangular meshes. A closer look reveals that the ZZ patch recovery is not superconvergent for
linear element under uniform triangular mesh of the Chevron pattern, nor it is superconvergent
for quadratic element at edge centers under uniform triangular mesh of the regular pattern
(see Section 4). This observation is confirmed by numerical tests (see Section 5). The question
naturally arises: Can we find a better recovery method? The new method should keep all
advantageous properties of the ZZ patch recovery while improving it under other situations,
e.g., the two cases we mentioned above.
∗This research was partially supported by the National Science Foundation grants DMS-0074301, DMS-
0079743, and DMS-0311807.†E-mail: [email protected]‡E-mail: [email protected]
1
In this paper, we introduce and analyze such a new gradient recovery method. Given a finite
element space of degree k, instead of fitting (in a least-squares sense) a polynomial of degree k
to gradient values at some sampling points on element patches (as in the ZZ patch recovery),
the new method fits a polynomial of degree k + 1 to solution values at some nodal points, and
then takes derivatives to obtain recovered gradient at each assembly points. The idea is also
related to the meshless method [12] where we only pay attention to nearby surrounding nodes
and not to elements. We shall prove that the new method is superconvergent for translation
invariant finite element spaces of any order. We shall also demonstrate that the new method
processes all known superconvergence and “ultraconvergence” (superconvergence with order
2) properties of the ZZ method, and is applicable to arbitrary grids with cost comparable to
the ZZ patch recovery. In computer implementation, there is no significant difference between
least-squares fitting a polynomial of degree k or degree k + 1, compared with the overall cost
in finite element solution.
The idea of fitting solution values was investigated earlier in [19] to recover finite element
solutions and to obtain the L2 norm a posteriori error estimates. Recently, Wang [18] proposed a
semi-local L2-projection (continuous least-squares fitting) to smooth the finite element solution.
Here we use the fitted solution values to recover the gradient and further to construct a posteriori
error estimates in the energy norm. Furthermore, there is no need for element patches in our
approach, and the method is “meshless”.
The application of the new recovery method to a posteriori error estimates and its compari-
son with the ZZ estimator will be discussed in a forthcoming paper, in which we shall utilize an
integral identity developed recently in [4] to prove the asymptotic exactness of the a posteriori
error estimator based on the new recovery method under arbitrary grid. In this respect, the
reader is also referred to a recent book by Ainsworth and Oden [1] for discussion of recovery
type a posteriori error estimators.
2 Meshless gradient recovery method
We introduce a new gradient recovery operator Gh : Sh → Sh × Sh, where Sh is a polynomial
finite element space of degree k over a triangulation Th. Given a finite element solution uh, we
need to define Ghuh at the following three types of nodes: vertices, edge nodes, and internal
nodes. For the linear element all nodes are vertices, for the quadratic element there are vertices
and edge-center nodes, and for the cubic element all three types of nodes are presented. After
determining values of Ghuh at all nodes, we obtain Ghuh ∈ Sh × Sh on the whole domain by
interpolation using the original nodal shape functions of Sh.
1) We start from vertices. For a vertex zzzi, let hi be the length of the longest edge attached
to zzzi. we select all nodes on the ball
Bhi(zzzi) = zzz ∈ D : |zzz − zzzi| ≤ hi,
where D is the solution domain. If the number of nodes n (including zzzi) is less than m =
(k + 2)(k + 3)/2, we go further and include nodes in B2hi(zzzi), continuing this process until we
have a sufficient number of nodes. We then denote them as zzzij , and fit a polynomial of degree
k + 1, in the least-squares sense, to the finite element solution uh at those nodes. Using the
2
local coordinates (x, y) with zzzi as the origin, the fitting polynomial is
pk+1(x, y;zzzi) = PPP Taaa = PPPTaaa,
with
PPP T = (1, x, y, x2, · · · , xk+1, xky, · · · , yk+1), PPPT
= (1, ξ, η, ξ2, · · · , ξk+1, ξkη, · · · , ηk+1);
aaaT = (a1, a2, · · · , am), aaaT = (a1, ha2, · · · , hk+1am),
where the scaling parameter h = hi. The coefficient vector aaa is determined by the linear system
AT Aaaa = ATbbbh, (2.1)
where bbbTh = (uh(zzzi1), uh(zzzi2), · · · , uh(zzzin)) and
A =
1 ξ1 η1 · · · ηk+11
1 ξ2 η2 · · · ηk+12
1 ξ3 η3 · · · ηk+13
......
......
...1 ξn ηn · · · ηk+1
n
.
The condition for (2.1) to have a unique solution is
RankA = m, (2.2)
which is almost always satisfied in practical situation when n ≥ m and grid points are reasonably
distributed. Now we define
Ghuh(zzzi) = ∇pk+1(0, 0;zzzi). (2.3)
2) If zzzi is an edge node which lies on an edge between two vertices zzzi1 and zzzi2 , we define
Ghuh(zzzi) = α∇pk+1(x1, y1;zzzi1) + (1 − α)∇pk+1(x2, y2;zzzi2), 0 < α < 1, (2.4)
where (x1, y1) (or (x2, y2)) is the local coordinates of zzzi with origin at zzzi1 (or zzzi2). The weight
α is determined by the ratio of the distances of zzzi to zzzi1 and zzzi2 .
3) If zzzi is an internal node which lies in a triangle formed by three vertices zzzi1 , zzzi2 , and zzzi3 ,
we define
Ghuh(zzzi) =3
∑
j=1
αj∇pk+1(xj , yj ;zzzij ),3
∑
j=1
αj = 1, αj > 0, (2.5)
where (xj , yj) is the local coordinates of zzzi with origin at zzzij . The weight αj is determined by
the ratio of the distances of zzzi to zzzi1 , zzzi2 , and zzzi3 .
In order to demonstrate the method, we shall discuss two examples in details. For the
sake of simplicity and superconvergence analysis, both examples are under uniform meshes.
Nevertheless, the method can be applied to arbitrary meshes even with curved boundaries, see
Fig. 15.
3
Example 1. Linear element on uniform triangular mesh. First, we consider the regular pattern
(Fig. 1). We fit a quadratic polynomial
p2(ξ, η) = (1, ξ, η, ξ2, ξη, η2)(a1, · · · , a6)T
in a least-squares sense with respect to the seven nodal values in (ξ, η) coordinates
~ξ = (0, 1, 0,−1,−1, 0, 1)T , ~η = (0, 0, 1, 1, 0,−1,−1)T .
Denote ~e = (1, 1, 1, 1, 1, 1, 1)T and set
A = (~e, ~ξ, ~η, ~ξ2, ~ξη, ~η2),
with ~ξ2 = (ξ21 , ξ
22 , · · · , ξ2
7)T , and ~ξη, ~η2 defined accordingly. We calculate
(AT A)−1AT =1
6
6 0 0 0 0 0 00 2 1 −1 −2 −1 10 1 2 1 −1 −2 −1−6 3 0 0 3 0 0−6 3 3 −3 3 3 −3−6 0 3 0 0 3 0
,
and obtain p2 from aaa = (AT A)−1ATbbb. In order to investigate the approximation property of
the recovery operator, we let bbbT = (u0, u1, . . . , u6) instead of using the finite element solution
uh. Recall
(a1, a2, a3, a4, a5, a6) = (a1, ha2, ha3, h2a4, h
2a5, h2a6),
and we obtain
p2(x, y) = u0 +1
6h[2(u1 − u4) + u2 − u3 + u6 − u5]x
+1
6h[2(u2 − u5) + u1 − u6 + u3 − u4]y +
1
2h2(u1 − 2u0 + u4)x
2
+1
2h2(u1 − 2u0 + u4 + u2 − u3 + u5 − u6)xy +
1
2h2(u2 − 2u0 + u5)y
2.
We see that
∂p2
∂x(x, y) =
1
6h[2(u1 − u4) + u2 − u3 + u6 − u5]
+1
h2(u1 − 2u0 + u4)x +
1
2h2(u1 − 2u0 + u4 + u2 − u3 + u5 − u6)y; (2.6)
∂p2
∂y(x, y) =
1
6h[2(u2 − u5) + u1 − u6 + u3 − u4]
+1
2h2(u1 − 2u0 + u4 + u2 − u3 + u5 − u6)x +
1
h2(u2 − 2u0 + u5)y. (2.7)
By the Taylor expansion, it is straight forward to verify that (2.6) and (2.7) provide a second
order approximation to ∇u, especially at (x, y) = (0, 0) where we have a finite difference scheme
1
6h
(
2(u1 − u4) + u2 − u3 + u6 − u5
2(u2 − u5) + u3 − u4 + u1 − u6
)
. (2.8)
4
We then obtain the recovered gradient at a vertex (see Fig. 1)
Ghu =1
6h
((
2
1
)
u1 +
(
1
2
)
u2 +
(
−1
1
)
u3 +
(
−2
−1
)
u4 +
(
−1
−2
)
u5 +
(
1
−1
)
u6
)
. (2.9)
With Ghu given at each vertex by (2.9), we are able to form a recovered gradient field by linear
interpolation using the finite element basis functions.
Next, we consider the Chevron mesh pattern. Following the same procedure as the above,
we obtain the recovered gradient at a vertex (see Fig. 2).
1
12h
(
6(u6 − u4)
−u1 − 4u2 − u3 + u4 − 2u5 + u6 + 6u7
)
. (2.10)
Again this is a second order approximation to the gradient.
Example 2. We consider quadratic element on uniform triangular mesh of the regular pattern.
We fit a cubic polynomial
p2(ξ, η) = (1, ξ, η, ξ2, . . . , ξη2, η3)(a1, · · · , a10)T
with respect to function values at nineteen nodes, which include seven vertices and twelve
edge centers (Fig. 5). Following the same procedure as in Example 1, we obtain the recovered
gradient at the vertex. We also obtain the recovered gradient at six edge centers by the averaging
procedure described in (2.4) with α = 1/2. We shall skip the detail and only demonstrate the
first components of the weights obtained from our new recovery procedure. Fig. 5 shows the
weights at the vertex. Figures 6, 9, and 11 show the weights at horizontal, vertical, and diagonal
edge centers, respectively, where the bottom picture is the average from the two on the top.
Each set of weights provides a finite difference scheme for the x-derivative. By the Taylor
expansion, we analyze the approximation quality of these finite difference schemes. This can
be done symbolically by Maple. We have found that they all have fourth order accuracy. We
list errors for the first component of the recovered gradient Ghu in approximating ∂xu for all
the previous four cases.
At a vertex:h4
960(8∂x∂4
yu + 16∂2x∂3
yu + 9∂3x∂2
yu + ∂4x∂yu + 2∂5
xu); (2.11)
at a horizontal edge center:
h4
68640(317∂x∂4
yu + 634∂2x∂3
yu + 1413∂3x∂2
yu + 1096∂4x∂yu + 6∂5
xu); (2.12)
at a vertical edge center:
h4
137280(75∂5
yu − 2248∂x∂4yu − 5264∂2
x∂3yu − 4294∂3
x∂2yu − 398∂4
x∂yu − 286∂5xu); (2.13)
and at a diagonal edge center:
h4
137280(75∂5
yu + 2623∂x∂4yu + 4478∂2
x∂3yu + 2740∂3
x∂2yu + 1765∂4
x∂yu + 1241∂5xu). (2.14)
5
We see that all finite difference schemes represented by weight stencils in Figures 5, 6, 9, and
11 produce the exact x-derivative for polynomials of degrees up to four. Without averaging, we
have third order accuracy at all edge centers instead of fourth order.
The recovered y-derivative can be determined in the same way. Again, with Ghu given
at each vertex and edge center, we are able to form a recovered gradient field by quadratic
interpolation using the finite element basis functions. By the approximation theory, Ghu−∇u
is of the third order.
We see from both examples that the recovery operator Gh provides a finite difference scheme
with k+1-order accuracy. Moreover, with averaging and uniform grid, Gh provides a k+2-order
accuracy at all mesh symmetry points including the vertex and edge centers for the quadratic
element. This is not by accident. In fact we have the following general theorems.
For the convenience of our analysis, we define an element patch ωi (around zzzi) which is a
union of elements that covers all nodes needed for the recovery of Ghuh(zzzi).
Theorem 2.1. The recovery operator Gh preserves polynomials of degree up to k + 1 for an
arbitrary grid. If zzzi is a mesh symmetry center of involved nodes and k = 2r, then Gh preserves
polynomials of degrees up to k + 2 at zzzi.
Proof. (i) When u ∈ Pk+1(ωi), the least squares fitting of a polynomial of degree k + 1 will
reproduce u, i.e., pk+1 = u on ωi. Therefore, Ghu(z) = ∇u(z) on ωi.
(ii) The value of the recovered gradient at each node zzzi can be equivalently expressed by a
finite difference scheme involving adjacent nodal values of u as following:
Ghu(zzzi) =1
h
∑
j
~cj(zzzi)u(zzzij),∑
j
~cj(zzzi) = ~0.
The key observation is that when nodes zzzij distribute symmetrically around zzzi, coefficients
~cj(zzzi) distribute anti-symmetrically. Furthermore, if k = 2r, and u is one of
(x − xi)k+2, (x − xi)
k+1(y − yi), (x − xi)k(y − yi)
2, . . . , (x − xi)(y − yi)k+1, (y − yi)
k+2,
which are all even functions with respect to zzzi = (xi, yi), then
Ghu(zzzi) =1
2h
∑
j
~cj(zzzi)(u(zzzij) − u(2zzzi − zzzij)) = 0 = ∇u(zzzi).
Note that u(zzzij) = u(2zzzi − zzzij) by symmetry since u is an even function with respect to zzzi.
(iii) When u ∈ Pk+2(ωi) (k = 2r) and nodes are symmetrically distributed around zzzi, using
(i), (ii), and the linear property of Gh, it is straightforward to derive Ghu(zzzi) = ∇u(zzzi).
Theorem 2.2. Let u ∈ W k+2∞ (ωi), then
‖∇u − Ghu‖L∞(ωi) ≤ Chk+1|u|W k+2∞ (ωi)
. (2.15)
If zzzi is a grid symmetry point and u ∈ W k+3∞ (ωi) with k = 2r, then
|(∇u − Ghu)(zzzi)| ≤ Chk+2|u|W k+3∞ (ωi)
. (2.16)
6
Proof. Recall the polynomial preserving property of Gh in Theorem 2.1, the conclusion follows
by applying the Hilbert-Bramble Lemma [5, 8].
Remark 2.3. As with the ZZ patch recovery, our new method provides a systematic way to
post-process (smooth) the finite element gradient. In addition, Theorem 2.1 and Theorem 2.2
reveal the most important property of the new recovery operator. In general, we are unable to
prove the same theorem for the ZZ patch recovery. Indeed, we shall demonstrate in Section 4
that the ZZ method has only O(h) recovery for the linear element (k = 1) on the uniform mesh
of the Chevron pattern and O(h2) recovery at the element edge center for the quadratic element
(k = 2) on the uniform mesh of the regular pattern.
Remark 2.4. As the ZZ patch recovery, our new method is also problem independent. In
addition, (2.15) is valid for arbitrary meshes as long as the rank condition (2.2) is satisfied.
Even (2.16) does not require uniform meshes as long as zzzi is a grid symmetry point.
2.1 Some practical issues
As we mentioned earlier, we need n ≥ m = (k + 2)(k + 3)/2 nodes for the least-squares fitting
to satisfy the rank condition (2.2). This requirement is usually satisfied within Bhi(zzzi). For an
interior vertex zzzi when using linear element, which needs at least six nodes to fit a quadratic
polynomial, there are only two exceptions: zzzi is linked to: a) three vertices (Fig. 13(a)), or
b) four vertices (Fig. 13(b)). When this happens, we can either include further all nodes in
Blhi(zzzi) for some integer l ≥ 2, or include only part of these nodes as described in Fig. 14(f-g).
The rule is to include enough nodes that make zzzi as centered as possible.
Recovering the gradient at a boundary vertex is more delicate, although there are many
ways to do that. One way is to use the strategy adopted for internal vertices as shown in
Fig. 14. However, our computational experiments indicated that this strategy is not efficient.
A more efficient one is depicted in Fig. 15. To recover the gradient at a vertex z ∈ ∂Ω, we look
for the nearest layer of vertices around z that contains at least one internal vertex. Let this
layer be the nth one, and denote the internal vertices in this layer by z1, . . . , zm where m ≥ 1.
The union of the sampling points used in recovering the gradient at z1, . . . , zm, and the mesh
nodes in the first n layers around z constitute the set of sampling points for recovering the
gradient at z.
With higher order elements, the task of selecting nodes can be conveniently carried out with
the concept of “element patch” as in ZZ patch recovery procedure. Attached to an interior
vertex, there are at least three triangles (see Fig. 13(a)). If we use these three triangles to form
an element patch, there are four vertices and six edges (with six edge centers). Function values
at these vertices and edge centers can uniquely determine a cubic polynomial by the least-
squares fitting. Therefore, selecting nodes for the quadratic finite element can be easily done
for an interior vertex with any geometric pattern. The cubic element has a similar situation
in which case we need at least fifteen nodes to fit a quartic polynomial. In fact, we can select,
it total, nineteen nodes on those three triangles in Fig. 13(a), with four vertices, twelve edge
nodes (two on each of the six edges), and three interior nodes. In general, the selected nodes
will be more local when the polynomial degree increases.
7
3 Superconvergence analysis
In this section, we utilize a tool in [14, 15, 17] to prove the superconvergence property of our
recovery operator. We refer readers to [5, 8] for general theory of the finite element method
and to [7, 9, 11, 17, 22] for the superconvergence theory.
First, we observe that the recovery operator results in a difference quotient. Let us take
linear element on uniform triangular mesh of the regular pattern as an example. The recovered
derivative at a nodal point O is (see Fig. 1)
∂hxu(O) =
1
6h[u2 − u3 + 2(u1 − u4) + u6 − u5].
Let φj be the nodal shape functions. Then we can express
∂hxu(O)φ0(x, y)
=1
6h[u2φ2(x, y + h) − u3φ3(x − h, y + h) + 2u1φ1(x + h, y)
− 2u4φ4(x − h, y) + u6φ6(x + h, y − h) − u5φ5(x, y − h)].
We see that the translations are in the directions of l1 = ±(1, 0), l2 = ±(0, 1), and l3 = ±(1,−1).
Therefore, we can express the recovered x-derivative as
∂hxuh(zzz) =
∑
|ν|≤M
3∑
i=1
C(i)ν,huh(zzz + νhli). (3.1)
The analysis here follows closely the argument of Wahlbin in [17, §8.2]. We consider the
finite element approximation of the solution of a scalar second order elliptic problem. With
D ⊂⊂ R2 a basic domain, Sh ⊂ H1(D) a parameterized family of finite element spaces, Ω ⊂⊂ D
and S0h(Ω) = v ∈ Sh : suppv ⊂ Ω, let u and uh ∈ Sh be two functions such that
A(u − uh, v) = 0, ∀v ∈ S0h(Ω),
where
A(w, v) =
∫ 2∑
i,j=1
aij∂w
∂xi
∂v
∂xj+
2∑
i=1
bi∂w
∂xiv + cwv.
Let Ω0 ⊂⊂ Ω1 ⊂⊂ Ω be separated by d ≥ c0h, let ` be a unit vector in R2, and let H be a
parameter, which is a constant times h. Denote by T `H , a translation by H in the direction `,
i.e., T `Hv(zzz) = v(zzz + H`), and for ν an integer,
T `νHv(zzz) = v(zzz + νH`) (3.2)
Then the finite element space is called translation invariant by H in the direction ` if
T `νHv ∈ S0
h(Ω), ∀v ∈ S0h(Ω1), |ν| ≤ M
for a fixed M . For constant coefficients A, we have
A(T `νH(u − uh), v) = A(u − uh, T `
−νHv) = A(u − uh, (T `νH)∗v) = 0, ∀v ∈ S0
h(Ω1).
8
Consequently, for Gh, a difference operator constructed from translations of type (3.2), we have
A(Gh(u − uh), vvv) = A(u − uh, G∗hvvv) = 0, ∀vvv ∈ S0
h(Ω1)2.
Therefore, from Theorem 5.5.2 of [17] (with F ≡ 0), we have
‖Gh(u − uh)‖L∞(Ω0) ≤ C(lnd
h)r min
vvv∈Sh×Sh
‖Ghu − vvv‖L∞(Ω1) (3.3)
+Cd−s−2/q‖Gh(u − uh)‖W−sq (Ω1)
. (3.4)
Here r = 1 for linear element and r = 0 for higher order elements. The first term on the right
hand side of (3.3) can be estimated by the standard approximation theory under the assumption
that the finite element space includes piecewise polynomials of degree k.
‖Ghu − vvv‖L∞(Ω1) ≤ Chk+1|u|W k+2∞ (Ω1). (3.5)
For the second term, we have
‖Gh(u − uh)‖W−sq (Ω1) = sup
φφφ∈C∞0 (Ω1)2,‖φφφ‖Ws
q′(Ω1)=1
(Gh(u − uh),φφφ).
Here
(Gh(u − uh),φφφ) = (u − uh, G∗hφφφ)
≤ C1‖u − uh‖L∞(Ω1+Mh)‖G∗hφφφ‖L1(Ω1+Mh)
≤ C2‖u − uh‖L∞(Ω1+Mh), (3.6)
where Ω1 + Mh define a sub-domain that stretches out Mh from Ω1. Note that when s ≥ 1,
‖G∗hφφφ‖L1(Ω1+Mh) is bounded uniformly with respect to h. Applying Theorem 5.5.2 in [17] again,
we have
‖u − uh‖L∞(Ω1+Mh) ≤ C(lnd
h)r min
v∈Sh
‖u − v‖L∞(Ω) + Cd−s−2/q‖u − uh‖W−sq (Ω)
≤ C(lnd
h)rhk+1‖u‖W k+1
∞ (Ω) + Cd−s−2/q‖u − uh‖W−sq (Ω). (3.7)
If the separation parameter d = O(1), then combining (3.3) to (3.7), we have shown:
‖Gh(u − uh)‖L∞(Ω0) ≤ C(ln1
h)rhk+1‖u‖W k+2
∞ (Ω) + C‖u − uh‖W−sq (Ω) (3.8)
Now we are ready for the main theorem of the paper.
Theorem 3.1. Let the coefficients in differential operator A be constants, let the finite element
space, which includes piecewise polynomials of degree k, be translation invariant in directions
required by the recovery operator Gh on Ω ⊂⊂ D, and let u ∈ W k+2∞ (Ω). Assume that A(u −
uh, v) = 0 for v ∈ S0h(Ω). Assume further that Theorem 5.5.2 in [17] is applicable. Then on
any interior region Ω0 ⊂⊂ Ω, there is a constant C independent of h and u such that
‖∇u − Ghuh‖L∞(Ω0) ≤ C(ln1
h)rhk+1‖u‖W k+2
∞ (Ω) + C‖u − uh‖W−sq (D), (3.9)
for some s ≥ 0 and q ≥ 1.
9
Proof. We decompose
∇u − Ghuh = (∇u − Ghu) + Gh(u − uh).
Then the conclusion follows by applying (2.15) of Theorem 2.1 to the first term and (3.8) to
the second term.
Remark 3.2. Theorem 3.1 is a superconvergence result under the condition
‖u − uh‖W−sq (D) ≤ Chk+σ, σ > 0.
For negative norm estimates, the reader is referred to [14].
The result is also quite general since it includes the most important cases such as linear and
quadratic elements, as well as higher-order elements. Based on (3.1), to utilize Theorem 3.1,
we have to use the same type of nodes to perform the recovery. This is not a restriction for
the regular pattern since all nodes are the same type. Unfortunately, it will be a restriction for
other patterns and higher order elements. In Fig. 16(a-c), we show sampling nodes of the other
three mesh patterns for the linear element; and in Fig. 16(d-e), we show sampling nodes of the
regular pattern for the quadratic element at a vertex as well as at a vertical edge center. Other
cases can be done similarly.
Although there is a restriction of selecting sampling points in using Theorem 3.1, our nu-
merical results indicate that the restriction is not necessary. Indeed, using the sampling nodes
demonstrated in Fig.s 1, 2, 5, 6, 9, 11, the following superconvergence results are observed.
1. Linear element (k = 1): superconvergence recovery is achieved for uniform triangular
meshes of all four patterns including the Chevron pattern.
2. Quadratic element (k = 2): Again superconvergence recovery is achieved for uniform
triangular meshes of all four patterns. In the literature, we know that the tangential derivative
of the finite element solution is superconvergent at two Gaussian points along the element edge
for certain uniform mesh patterns [2, 3, 17]. Zienkiewicz-Zhu reported in 1992 [23] that their
method produced O(h4) gradient recovery at element vertices for the uniform triangular mesh
of the regular pattern. However, since the ZZ patch recovery only results in O(h2) recovery at
element edge centers, it does not generate superconvergence recovery for the quadratic element
on the whole patch. We shall see from our numerical examples in Section 5 that our new recovery
produces O(h4) gradient recovery at element vertices as well as at element edge centers. By
quadratic interpolation at vertices and element edge centers, this will surely result in an O(h3)
recovery.
Remark 3.3. Following the argument in [17, §8.4], the result of Theorem 3.1 can be generalized
to variable coefficients cases.
Remark 3.4. By constructing tensor-product of smoothest B-splines, Bramble-Schatz [6] de-
signed a local averaging method (K-operator) to achieve superconvergent approximation to
solution values. The argument was extended to include superconvergent approximation to any
depuratives of the solution [16]. However, the method requires meshes to be locally translation
invariant in all of the axes directions. For this reason, the method has not been implemented
in commercial finite element codes.
10
In comparison, our new recovery method as well as the ZZ patch recovery method are
applicable to arbitrary meshes in practice, even though we have only proved superconvergence
under translation invariant meshes. Currently, the ZZ error estimator based on the patch
recovery technique is used in many commercial codes, such as ANSYS, MCS/NASTRAN-Marc,
Pro/MECHANICA (a product of Parametric Technology), and I-DEAS (product of SDRC,
part of EDS), for the purpose of smoothing and adaptive remeshing. It is also used in NASA’s
COMET-AR (COmputational MEchanics Testbed With Adaptive Refinement). We hope that
our new method can find its application in practical engineering computation.
Remark 3.5. Another important feature of the new method shared with the ZZ patch recovery
is its problem independence. Although we have only proven superconvergence for second-order
elliptic problems, the recovery procedure can be applied to many other problems, including
nonlinear problems.
Finally, we want to explain the gap between the general setup (2.3)–(2.5) and the special
case (3.1). Technically, we have only proven superconvergence for this special case. When
meshes are distorted, the theoretical approach differs. We shall use the decomposition
∇u − Ghuh = (∇u − GhuI) + Gh(uI − uh),
where uI is any interpolation (of u) in the finite element space. The first term on the right hand
side is superconvergent for any mesh by the polynomial preserving property. Under some mild
mesh conditions, the recovery operator Gh is bounded in the sense |Ghv(z)| ≤ C‖∇v‖L2(ωz) for
any v in the finite element space. Therefore, the error bound of ∇u − Ghuh is linked with the
error bound of ∇(uI −uh). It is possible to show that ‖∇(uI −uh)‖L2(Ω) is of the order O(h1+ρ)
with ρ ∈ (0, 1) for the linear element under a mildly structured mesh, such as a Delaunay type
mesh. We see that, for distorted meshes, we lose full order superconvergence. However, as long
as there is some mesh structure, we are still able to maintain superconvergence of order ρ > 0.
This observation is confirmed by numerical tests and two examples will be demonstrated in
Section 5. For more details along the line of unstructured meshes, the reader is referred to [13]
and [20].
4 Comparison with the ZZ patch recovery
As we proved in Theorem 2.1, the new recovery method is degree k + 1 polynomial preserving
for finite element method of degree k. In general, the ZZ patch recovery is not degree k + 1
polynomial preserving even for uniform meshes. To illustrate this, we consider quadratic element
on uniform triangular grid of the regular pattern. As a comparison, we display in Figures 7, 8,
10, and 12 the first components of ~cZZj (zzzi) obtained from the ZZ patch recovery at a vertex zzz0, a
horizontal edge center zzz1, a vertical edge center zzz2, and a diagonal edge center zzz3, respectively.
It is straightforward to verify that the finite difference scheme represented by the stencils in
Figures 6, 9, and 11 produces the exact x-derivative for polynomials of degrees up to four, while
the the stencils in Figures 8, 10, and 12 can only produce the exact x-derivative for polynomials
of degrees up to two. We list errors for the first component of the recovered gradient from the
ZZ method in approximating ∂xu for all different cases in Figures 5, 6, 9, and 11.
11
At a vertex:
h4
1920(10∂x∂4
yu + 20∂2x∂3
yu + 15∂3x∂2
yu + 5∂4x∂yu + 4∂5
xu); (4.1)
at a horizontal edge center:h2
264∂3
xu; (4.2)
at a vertical edge center:h2
264(3∂2
x∂yu + 2∂3xu); (4.3)
at a diagonal edge center:h2
264(3∂2
x∂yu + ∂3xu). (4.4)
We see that only second order accuracy is achieved at all edge centers even with averaging.
On an irregular grid, the ZZ patch recovery does not reproduce a cubic polynomial even at
the vertex. When we distort the central node in an element patch of the regular pattern by δh
in both x and y directions (see Fig. 17), the convergence rate drops from four to two as we can
see from the following equation:
δ2h2
120(11δ4 + 50δ2 + 44)[(533 − 454δ2 + 26δ4)∂3
xu + (829 − 1409δ2 − 218δ4)∂2x∂yu
+ (275 − 1483δ2 − 514δ4)∂x∂2yu + (11 + 34δ2 + 18δ2)∂3
yu].
We can show that for linear element under uniform triangular mesh of the regular pattern,
the new method is the same as the ZZ patch recovery as well as the weighted average. In
other words, all three methods produce the same recovery operator Gh. We can further show
that under the uniform triangular mesh of the Union Jack and the Criss Cross patterns, our
procedure is equivalent to the ZZ patch recovery and the weighted average for k = 1, i.e.,
all three recovery techniques produce O(h2) recovery for linear element under the uniform
triangular meshes of the Union Jack the Criss Cross patterns.
However, for irregular grids, the new method produces the exact gradient for polynomials
of degrees up to 2 while the other two methods can only maintain polynomials of degree 1 for
linear finite elements. This is even the case with the uniform mesh of the Chevron pattern.
In Fig. 3 and Fig. 4, we plot the stencils for the weighted average and the ZZ patch recovery,
respectively. It is straightforward to verify that both of them result in only a first order recovery
at the center, comparing with the second order scheme of Fig. 2. In the last section, we shall
demonstrate that our new method indeed results in a superconvergence gradient recovery at
each interior vertex.
The discussion in this section concerns polynomial preserving properties of three different
recovery operators. Nevertheless, this property is only one aspect of the recovery operator
since it does not involve the finite element solution. Even without the polynomial preserving
property, the ZZ patch recovery is very effective for the linear element in Delaunay type meshes.
The theoretical reason was explained in [20].
12
5 Numerical tests
In this section, three test problems are used to verify superconvergence and ultraconvergence
of our new gradient recovery method. We shall especially demonstrate the superiority of the
new method over the ZZ patch recovery by comparing the two under 1) linear element on the
uniform grid of the Chevron pattern; and 2) quadratic element on the uniform grid of the regular
pattern. In order to exclude the boundary singularity, both of our test cases have analytic exact
solutions.
Case 1. Our first example is a test case in [23], the Poisson equation with zero boundary
condition on the unit square with the exact solution
u(x, y) = x(1 − x)y(1 − y).
Case 2. Our second example is
−∆u = 2π2 sin πx sin πy in Ω = [0, 1]2, u = 0 on ∂Ω.
The exact solution is u(x, y) = sin πx sin πy.
Case 3. Our final example is again the Poisson equation. However, this time Ω = (−1, 1)2 \
[1/2, 1)2 is the L-shaped domain. In addition, the boundary condition is no longer homogeneous.
Using a polar coordinate system at (1/2,1/2), the solution can be expressed as
u = r13 sin
2θ − π
3, π/2 ≤ θ ≤ 2π.
The initial mesh for linear element is obtained by decomposing the unit square into 4 × 4
uniform squares and dividing each sub-square into two triangles with the Chevron pattern.
Computation is performed on four different mesh levels based on bisection refinement. We
define ‖ · ‖∞,N as a discrete maximum norm at all nodal points in an interior region [1/8, 7/8]2.
Fig. 18 and Fig. 19 compare the performance of the new recovery and the ZZ patch recovery.
They show a second-order convergent rate (a superconvergence result) of the recovered gradient
by our new method in both test cases while only a first-order convergent rate for the ZZ patch
recovery.
The quadratic element starts with the initial mesh of the regular pattern with the same
amount of elements as in the linear case. However, in order to maintain the edge centers we
use tri-section, i.e., 3 × 3 refinement to obtain the next two mesh levels (with 2(12 × 12) and
2(36× 36) elements, respectively). We define ‖ · ‖∞,Nv and ‖ · ‖∞,Ne as two discrete maximum
norms at all vertices and edge centers, respectively, in an interior region [1/9, 8/9]2. Fig. 20
indicates a six-order convergent rate (a surprising result!) of the recovered gradient by our new
method for Case 1 in both discrete norms and shows only a second-order convergent rate for the
ZZ patch recovery at the edge centers. Similarly, Fig. 18 indicates a fourth-order convergent
rate (an ultraconvergence result) of the recovered gradient by our new method for Case 2 in
both discrete norms and shows only a second-order convergent rate for the ZZ patch recovery
at the edge centers.
Next, we consider unstructured meshes. For the model problem in Case 2, the initial mesh
(Fig. 23) is obtained by the Delaunay triangulation. In successive iterations, the mesh triangles
13
are refined regularly (by linking edge centers). To distinguish the behavior inside and near
the boundary, we decompose the domain into Ω1 and Ω2, where Ω2 is a boundary layer of
width about 1/8 and Ω1 = Ω \ Ω2. As we can see in Fig. 25, the rate of convergence for the
recovered gradient in the L2-norm is about 3 in Ω1 and slightly less than 3 in Ω2, which implies
a superconvergent recovery. In addition, this figure clearly indicates that the new recovery is
more accurate than the ZZ patch recovery.
In Case 3, the solution has a corner singularity at (1/2, 1/2). In order to control the pollution
error, a local mesh refinement is applied to the initial mesh near this corner as shown in Fig. 22.
Again, we decompose Ω into Ω1 and Ω2 as before in order to isolate the boundary behavior.
The numerical results are depicted in Fig. 24. Due to the corner singularity, the recovered
gradient has a lower convergence rate, especially near the boundary, for both the ZZ and the
new recoveries. However, inside the domain, both methods achieve superconvergence. In this
case, the two methods are comparable.
As we mentioned earlier in Section 4, for the linear element, the new recovery method
is the same as the ZZ patch recovery (and the weighted average) for the uniform triangular
mesh of the regular patter, and is equivalent to the ZZ patch recovery (as well as the weighted
average) under the uniform triangular mesh of the Criss Cross and the Union Jack patterns.
Therefore, the new method inherits the superconvergence property of the ZZ patch recovery
under these situations. Our theoretical and numerical results show that the new method also
provides superconvergent recovery for the Chevron pattern, which is a significant improvement
over the ZZ patch recovery. As for the quadratic element, the new method not only keeps the
ultraconvergence of the ZZ patch recovery at the vertices but also produces ultraconvergent
recovery at element edge centers, thereby providing a superconvergent recovery on the whole
interior domain by interpolation using the quadratic finite element basis functions. This is also
a significant improvement over the ZZ patch recovery.
In summary, the new recovery method keeps all known superconvergent properties of the
ZZ patch recovery while out-performing it in the cases of quadratic element at edge centers and
linear element for the Chevron mesh. Our further investigation will be devoted to analysis of the
new recovery method in application to a posteriori error estimates, especially under irregular
meshes.
Acknowledgement. The first author would like to thank Dr. J.Z. Zhu, one of the inventors
of the ZZ patch recovery method, for his encouragement and valuable discussions on the research
in this direction.
14
References
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University Press, London, 2001.
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Springer-Verlag, New York, 1994.
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element method, Math. Comp., 31 (1977), pp. 74–111.
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vergence, Post-processing, and A Posteriori Estimates, Lecture Notes in Pure and Applied
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linear and bilinear finite elements, Numer. Meth. PDEs, 15 (1999), pp. 151–167.
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nese), Hebei University Press, P.R. China, 1996.
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puter Methods in Applied Mechanics and Engineering, 139 (1996).
[13] A. Naga and Z. Zhang, A posteriori error estimates based on polynomial preserving recov-
ery, Accepted for publication by SIAM J. Numer. Anal.
[14] J.A. Nitsche and A.H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp.,
28 (1974), pp. 937–958.
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ods, Part II, Math. Comp., 64 (1995), pp. 907–928.
[16] V. Thomee, High order local approximation to derivatives in the finite element method,
Math. Comp., 34 (1977), pp. 652–660.
15
[17] L.B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in
Mathematics, Vol.1605, Springer, Berlin, 1995.
[18] J. Wang, A superconvergence analysis for finite element solutions by the least-squares
surface fitting on irregular meshes for smooth problems, J. Math. Study, 33-3 (2000).
[19] N.-E. Wiberg and X.D. Li, Superconvergence patch recovery of finite element solutions and
a posteriori L2 norm error estimate, Commun. Num. Meth. Eng., 37 (1994), pp. 313–320.
[20] J. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimators for mildly
structured grids, Accepted for publication by Math. Comp.
[21] Z. Zhang, Ultraconvergence of the patch recovery technique II, Math. Comp., 69 (2000),
pp. 141–158.
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Science Press, China, 1989 (in Chinese).
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1331–1364.
16
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@@
(
00
)
(
21
)
(
12
)(
−11
)
(
−2−1
)
(
−1−2
) (
1−1
)
Fig. 1. New Recovery: Denominator 6h
@@
@@
@@
@@
@@
@@
@@
@@
(
0−2
)
(
61
)(
−61
)
(
0−1
)(
0−1
)
(
06
)
(
0−4
)
Fig. 2. New Recovery: Denominator 12h
@@
@@
@@
@@
@@
@@
@@
@@
(
00
)
(
21
)(
−21
)
(
1−1
)(
−1−1
)
(
02
)
(
0−2
)
Fig. 3. Weighted Average: Denominator 6h
@@
@@
@@
@@
@@
@@
@@
@@
(
0−6
)
(
62
)(
−62
)
(
1−2
)(
−1−2
)
(
08
)
(
0−2
)
Fig. 4. ZZ Recovery: Denominator 14h
17
4 −4
8 0 −8
4 −4
−7 −11
−22
11
0
7
22
−7 −11 11
0
7
Fig. 5. New Recovery: Denominator 30h
530 46
508 −1248 2708
530 46
−8 −1084
−1172
−644
−426
1136
104
−8 −1084 −644
−426
1136
−46 −530
−2708 1248 −508
−46 −530
−1136 644
−104
1084
426
8
1172
−1136 644 1084
426
8
265 0 −265
254 −1978 1978 −254
265 0 −265
−4 −542
−586
−890
−213
890
0
542
213
4
586
−4 −542 −890
−213
890 542
213
4
Fig. 6. New Recovery: Denominator 4290h
1 −1
2 0 −2
1 −1
−1 −5
−10
5
0
1
10
−1 −5 5
0
1
1 2
−1 0 1
−2 −1
0 −5
5
−10
−1
−1
−5
1 10 5
1
0
Fig. 7. ZZ Recovery: Denominator 12h
104 −16
56 0 296
104 −16
26 −206
−236
−146
−88
326
−116
26 −206 −146
−88
326
16 −104
−296 0 −56
16 −104
−326 146
116
206
88
−26
236
−326 146 206
88
−26
52 0 −52
28 −148 148 −28
52 0 −52
13 −103
−118
−236
−44
236
0
103
44
−13
118
13 −103 −236
−44
236 103
44
−13
Fig. 8. ZZ Recovery: Denominator 660h
18
−416 2708
2708 −1248 508
508 −1648
−2142
−712 −4406
−4186
104 4414
3118
−1306 −1172 2018
1290
−140
1648 −508
−508 1248 −2708
−2708 416
140 −2018
−3118
1172
−1290
1306
4186
−4414 −104 4406
2142
712
824 −254
−462 1978 −1354
1354 −1978 462
254 −824
70 −1009
−2630
586
−645
653
2093
−356 −4410
−2093
0 4410
2630
356
−653 −586 1009
645
−70
Fig. 9. New Recovery: Denominator 8250h
−44 148
148 0 28
28 −44
−104
−14 −358
−388
−58 430
212
10 −118 182
16
−74
44 −28
−28 0 −148
−148 44
74 −182
−212
118
−16
−10
388
−430 58 358
104
14
22 −14
−36 74 −74
74 −74 36
14 −22
37 −91
−158
59
−8
−5
194
−7 −394
−194
0 394
158
7
5 −59 91
8
−37
Fig. 10. ZZ Recovery: Denominator 660h
19
−2708 416
−508 1248 −2708
1648 −508
2142
−4414 −104
−3118
4406 712
4186
140 −2018 1172
−1290
1306
508 −1648
2708 −1248 508
−416 2708
−1306 −1172
−4186
2018
1290
−140
3118
−712 −4406 104
−2142
4414
254 −824
1354 −1978 462
−462 1978 −1354
824 −254
−653 −586
−2093
1009
645
−70
2630
−356 −4410 0
−2630
4410 356
2093
70 −1009 586
−645
653
Fig. 11. New Recovery: Denominator 8250h
−148 44
−28 0 −148
44 −28
104
−430 58
−212
358 14
388
74 −182 118
−16
−10
28 −44
148 0 28
−44 148
10 −118
−388
182
16
−74
212
−14 −358 −58
−104
430
14 −22
74 −74 36
−36 74 −74
22 −14
5 −59
−194
91
8
−37
158
−7 −394 0
−158
394 7
194
37 −91 59
−8
−5
Fig. 12. ZZ Recovery: Denominator 660h
20
b a
Fig. 13. Sampling points selection: Interior
∂Ω
a b
∂Ω
c
∂Ω
d
∂Ω
e
∂Ω
Fig. 14. Sampling points selection: boundary strategy 1
Fig. 15. Sampling points selection: boundary strategy 2
21
d
b c a
e
Fig. 16. Nodes selections that satisfy conditions in Theorem 3.1
(δ,δ)
(0,0)
Fig. 17. Distorted regular pattern
Fig. 18. Linear element Case 1 (Chevron) Fig. 19. Linear element Case 2 (Chevron)
22
Fig. 20. Quadratic element Case 1 (Regular pattern)
Fig. 21. Quadratic element case 2 (Regular pattern)
23